Relational Algebra Lecture 2 Relational Model • • • • • • • • Basic Notions Fundamental Relational Algebra Operations Additional Relational Algebra Operations Extended Relational Algebra Operations Null Values Modification of the Database Views Bags and Bag operations Basic Structure • Formally, given sets D1, D2, …. Dn a relation r is a subset of D1 x D2 x … x Dn Thus, a relation is a set of n-tuples (a1, a2, …, an) where each ai Di • Example: customer_name = {Jones, Smith, Curry, Lindsay} customer_street = {Main, North, Park} customer_city = {Harrison, Rye, Pittsfield} Then r = { (Jones, Main, Harrison), (Smith, North, Rye), (Curry, North, Rye), (Lindsay, Park, Pittsfield) } is a relation over customer_name , customer_street, customer_city Attribute Types • Each attribute of a relation has a name • The set of allowed values for each attribute is called the domain of the attribute • Attribute values are (normally) required to be atomic; that is, indivisible – Note: multivalued attribute values are not atomic ({secretary. clerk}) is example of multivalued attribute position – Note: composite attribute values are not atomic • The special value null is a member of every domain • The null value causes complications in the definition of many operations – We shall ignore the effect of null values in our main presentation and consider their effect later Relation Schema • A1, A2, …, An are attributes • R = (A1, A2, …, An ) is a relation schema Example: Customer_schema = (customer_name, customer_street, customer_city) • r(R) is a relation on the relation schema R Example: customer (Customer_schema) Relation Instance • The current values (relation instance) of a relation are specified by a table • An element t of r is a tuple, represented by a row in a attributes table customer_name customer_street customer_city Jones Smith Curry Lindsay Harrison Rye Rye Pittsfield Main North North Park customer (or columns) tuples (or rows) Database • A database consists of multiple relations • Information about an enterprise is broken up into parts, with each relation storing one part of the information account : stores information about accounts depositor : stores information about which customer owns which account customer : stores information about customers • Storing all information as a single relation such as bank(account_number, balance, customer_name, ..) results in repetition of information (e.g., two customers own an account) and the need for null values (e.g., represent a customer without an account) Query Languages • Language in which user requests information from the database. • Categories of languages – Procedural – Non-procedural, or declarative • “Pure” Procedural languages: – Relational algebra – Tuple relational calculus – Domain relational calculus • Pure languages form underlying basis of query languages that people use. What is “algebra” • Mathematical model consisting of: – Operands --- Variables or values; – Operators --- Symbols denoting procedures that construct new values from a given values • Relational Algebra is algebra whose operands are relations and operators are designed to do the most commons things that we need to do with relations Basic Relational Algebra Operations • • • • • Select Project Union Set Difference (or Substract or minus) Cartesian Product Select Operation • Notation: p(r) • p is called the selection predicate • Defined as: p(r) = {t | t r and p(t)} Where p is a formula in propositional calculus consisting of terms connected by : (and), (or), (not) Each term is one of: <attribute>op <attribute> or <constant> where op is one of: =, , >, . <. • Example of selection: Account(account_number, branch_name,balance) branch-name=“Perryridge”(account) Select Operation – Example • Relation r • A=B ^ D > 5 (r) A B C D 1 7 5 7 12 3 23 10 A B C D 1 7 23 10 Project Operation • Notation: A1, A2, …, Ak (r) where A1, A2 are attribute names and r is a relation. • The result is defined as the relation of k columns obtained by erasing the columns that are not listed • Duplicate rows removed from result, since relations are sets • E.g. to eliminate the branch-name attribute of account account-number, balance (account) • If relation Account contains 50 tuples, how many tuples contains account-number, balance (account) ? • If relation Account contains 50 tuples, how many tuples contains , balance (account) ? Project Operation – Example • Relation r: • A,C (r) A A B C 10 1 20 1 30 1 40 2 C 1 1 1 2 A = C 1 1 2 That is, the projection of a relation on a set of attributes is a set of tuples Union Operation • Consider relational schemas: Depositor(customer_name, account_number) Borrower(customer_name, loan_number) • For r s to be valid. 1. r, s must have the same number of attributes 2. The attribute domains must be compatible (e.g., 2nd column of r deals with the same type of values as does the 2nd column of s) Find all customers with either an account or a loan customer-name (depositor) customer-name (borrower) Union Operation • Notation: r s • Defined as: r s = {t | t r or t s} Union Operation – Example • Relations r, s: A B A B 1 2 2 3 1 s r r s: A B 1 2 1 3 Set Difference Operation • Notation r – s • Defined as: r – s = {t | t r and t s} • Set differences must be taken between compatible relations. – r and s must have the same number of attributes – attribute domains of r and s must be compatible Set Difference Operation – Example • Relations r, s: A B A B 1 2 2 3 1 s r r – s: A B 1 1 Cartesian-Product Operation • Notation r x s • Defined as: r x s = {t q | t r and q s} • Assume that attributes of r(R) and s(S) are disjoint. (That is, R S = ). • If attributes of r(R) and s(S) are not disjoint, then renaming must be used. Cartesian-Product Operation-Example Relations r, s: A B C D E 1 2 10 10 20 10 a a b b r s r x s: A B C D E 1 1 1 1 2 2 2 2 10 10 20 10 10 10 20 10 a a b b a a b b Composition of Operations • Can build expressions using multiple operations • Example: A=C(r s) • rs • A=C(r s) A B C D E 1 1 1 1 2 2 2 2 10 10 20 10 10 10 20 10 a a b b a a b b A B C D E 1 2 2 10 20 20 a a b Rename Operation • Allows us to name, and therefore to refer to, the results of relational-algebra expressions. • Allows us to refer to a relation by more than one name. Example: X (E) returns the expression E under the name X If a relational-algebra expression E has arity n, then (A1, A2, …, An) (E) xx E under the name X, and with returns the result of expression the attributes renamed to A1, A2, …., An. Banking Example branch (branch-name, branch-city, assets) customer (customer-name, customer-street, customercity) account (account-number, branch-name, balance) loan (loan-number, branch-name, amount) depositor (customer-name, account-number) borrower (customer-name, loan-number) Keys • Let K R • K is a superkey of R if values for K are sufficient to identify a unique tuple of each possible relation r(R) – by “possible r ” we mean a relation r that could exist in the enterprise we are modeling. – Example: {customer_name, customer_street} and {customer_name} are both superkeys of Customer, if no two customers can possibly have the same name. • K is a candidate key if K is minimal Example: {customer_name} is a candidate key for. • Primary Key Keys Superkeys Candidate keys K Primary key Example Queries • Find all loans of over $1200 amount > 1200 (loan) • Find the loan number for each loan of an amount greater than $1200 loan-number (amount > 1200 (loan)) Example Queries • Find the names of all customers who have a loan, an account, or both, from the bank customer-name (borrower) customer-name (depositor) • Find the names of all customers who have a loan and an account at bank. customer-name (borrower) customer-name (depositor) Additional Operations We define additional operations that do not add any power to the relational algebra, but that simplify common queries. • Set intersection • Natural join • Division • Assignment Set-Intersection Operation Notation: r s Defined as: r s ={ t | t r and t s } Assume: – r, s have the same arity – attributes of r and s are compatible • Note: r s = r - (r - s) • • • • Set-Intersection Operation - Example • Relation r, s: A B 1 2 1 r • rs A B 2 3 s A B 2 Natural-Join Operation Notation: r s • Let r and s be relations on schemas R and S respectively. Then, r s is a relation on schema R S obtained as follows: – Consider each pair of tuples tr from r and ts from s. – If tr and ts have the same value on each of the attributes in R S, add a tuple t to the result, where • t has the same value as tr on r • t has the same value as ts on s • Example: R = (A, B, C, D) S = (E, B, D) – Result schema = (A, B, C, D, E) – r s is defined as: r.A, r.B, r.C, r.D, s.E (r.B = s.B r.D = s.D (r x s)) Natural Join Operation – Example • Relations r, s: A B C D B D E 1 2 4 1 2 a a b a b 1 3 1 2 3 a a a b b r r s s A B C D E 1 1 1 1 2 a a a a b Natural Join Example sid bid day 22 58 101 103 10/ 1 0/ 9 6 11/ 1 2/ 9 6 sid sn am e rat ing ag e 22 dus tin 7 45.0 31 lub b er 8 55.5 58 rusty 10 35.0 R1 R1 S1 S1 = sid sname rating age bid day 22 58 dustin rusty 7 10 45.0 35.0 101 103 10/10/96 11/12/96 Other Types of Joins • Condition Join (or “theta-join”): R c S c ( R S ) • Result schema same as that of crossproduct. • May have fewer tuples than cross-product. • Equi-Join: Special case: condition c contains only conjunction of equalities. “Theta” Join Example sid bid day sid sname rating age 22 58 101 103 10/10/96 11/12/96 22 dustin 7 45.0 31 58 lubber rusty 8 10 55.5 35.0 R1 S1 S1 S1.sid R1.sid R1 = (sid) sname rating age (sid) bid day 22 31 dustin lubber 7 8 45.0 55.5 58 58 103 103 11/12/96 11/12/96 Division Operation Notation: rs • Suited to queries that include the phrase “for all”. • Let r and s be relations on schemas R and S respectively where – R = (A1, …, Am, B1, …, Bn) – S = (B1, …, Bn) The result of r s is a relation on schema R – S = (A1, …, Am) r s = { t | t R-S(r) u s ( tu r ) } Division Operation – Example Relations r, s: r s: A A B B 1 2 3 1 1 1 3 4 6 1 2 1 r 2 s Another Division Example Relations r, s: A B C D E D E a a a a a a a a a a b a b a b b 1 1 1 1 3 1 1 1 a b 1 1 r r s: A B C a a s Division Operation • Definition in terms of the basic algebra operation Let r(R) and s(S) be relations, and let S R r s = R-S (r) –R-S ( (R-S (r) x s) – R-S,S(r)) To see why – R-S,S(r) simply reorders attributes of r – R-S(R-S (r) x s) – R-S,S(r)) gives those tuples t in R-S (r) such that for some tuple u s, tu r. Assignment Operation • The assignment operation () provides a convenient way to express complex queries. – Write query as a sequential program consisting of • a series of assignments • followed by an expression whose value is displayed as a result of the query. – Assignment must always be made to a temporary relation variable. • Example: Write r s as temp1 R-S (r) temp2 R-S ((temp1 x s) – R-S,S (r)) result = temp1 – temp2 Extended Relational-Algebra-Operations • Generalized Projection • Outer Join • Aggregate Functions Generalized Projection • Extends the projection operation by allowing arithmetic functions to be used in the projection list. F1, F2, …, Fn(E) • E is any relational-algebra expression • Each of F1, F2, …, Fn are are arithmetic expressions involving constants and attributes in the schema of E. • Given relation credit-info(customer-name, limit, creditbalance), find how much more each person can spend: customer-name, limit – credit-balance (credit-info) Aggregate Functions and Operations • Aggregation function takes a collection of values and returns a single value as a result. avg: average value min: minimum value max: maximum value sum: sum of values count: number of values • Aggregate operation in relational algebra G1, G2, …, Gn g F1( A1), F2( A2),…, Fn( An) (E) – E is any relational-algebra expression – G1, G2 …, Gn is a list of attributes on which to group (can be empty) – Each Fi is an aggregate function – Each Ai is an attribute name Aggregate Operation – Example • Relation r: g sum(c) (r) A B C 7 sum-C 27 7 3 10 Aggregate Operation – Example • Relation account grouped by branch-name: branch-name account-number Perryridge Perryridge Brighton Brighton Redwood branch-name g balance A-102 A-201 A-217 A-215 A-222 sum(balance) 400 900 750 750 700 (account) branch-name Perryridge Brighton Redwood balance 1300 1500 700 Aggregate Functions • Result of aggregation does not have a name – Can use rename operation to give it a name – For convenience, we permit renaming as part of aggregate operation branch-name g sum(balance) as sum-balance (account) Outer Join – Example • Relation loan loan-number branch-name L-170 L-230 L-260 Downtown Redwood Perryridge amount 3000 4000 1700 Relation borrower customer-name loan-number Jones Smith Hayes L-170 L-230 L-155 Outer Join • An extension of the join operation that avoids loss of information. • Computes the join and then adds tuples form one relation that does not match tuples in the other relation to the result of the join. • Uses null values: – null signifies that the value is unknown or does not exist – All comparisons involving null are (roughly speaking) false by definition. • We shall study precise meaning of comparisons with nulls later Left Outer Join • Join loan Borrower loan-number L-170 L-230 branch-name Downtown Redwood amount customer-name 3000 4000 Jones Smith amount customer-name Left Outer Join loan Borrower loan-number L-170 L-230 L-260 branch-name Downtown Redwood Perryridge 3000 4000 1700 Jones Smith null Right Outer Join, Full Outer Join • Right Outer Join loan borrower loan-number L-170 L-230 L-155 branch-name Downtown Redwood null amount 3000 4000 null customer-name Jones Smith Hayes Outer Join loan borrower loan-number L-170 L-230 L-260 L-155 branch-name Downtown Redwood Perryridge null amount 3000 4000 1700 null customer-name Jones Smith null Hayes Null Values • It is possible for tuples to have a null value, denoted by null, for some of their attributes • null signifies an unknown value or that a value does not exist. • The result of any arithmetic expression involving null is null. • Aggregate functions simply ignore null values • For duplicate elimination and grouping, null is treated like any other value, and two nulls are assumed to be the same Null Values • Comparisons with null values return the special truth value unknown – If false was used instead of unknown, then not (A < 5) would not be equivalent to A >= 5 • Three-valued logic using the truth value unknown: – OR: (unknown or true) = true, (unknown or false) = unknown (unknown or unknown) = unknown – AND: (true and unknown) = unknown, (false and unknown) = false, (unknown and unknown) = unknown – NOT: (not unknown) = unknown • Result of select predicate is treated as false if it evaluates to unknown Modification of the Database • The content of the database may be modified using the following operations: – Deletion – Insertion – Updating • All these operations are expressed using the assignment operator. Deletion • A delete request is expressed similarly to a query, except instead of displaying tuples to the user, the selected tuples are removed from the database. • Can delete only whole tuples; cannot delete values on only particular attributes • A deletion is expressed in relational algebra by: rr–E where r is a relation and E is a relational algebra query. Deletion Examples • Delete all account records in the Perryridge branch. account account – branch_name = “Perryridge” (account ) • Delete all loan records with amount in the range of 0 to 50 loan loan – amount 0 and amount 50 (loan) Delete all accounts at branches located in Needham. r1 branch_city = “Needham” (account branch ) r2 branch_name, account_number, balance (r1) r3 customer_name, account_number (r2 account account – r2 depositor depositor – r3 depositor) Insertion • To insert data into a relation, we either: – specify a tuple to be inserted – write a query whose result is a set of tuples to be inserted • in relational algebra, an insertion is expressed by: r r E where r is a relation and E is a relational algebra expression. • The insertion of a single tuple is expressed by letting E be a constant relation containing one tuple. Insertion Examples • Insert information in the database specifying that Smith has $1200 in account A-973 at the Perryridge branch. account account {(“Perryridge”, A-973, 1200)} depositor depositor {(“Smith”, A-973)} Provide as a gift for all loan customers in the Perryridge branch, a $200 savings account. Let the loan number serve as the account number for the new savings account. r1 (branch_name = “Perryridge” (borrower loan)) account account branch_name, loan_number,200 (r1) depositor depositor customer_name, loan_number (r1) Updating • A mechanism to change a value in a tuple without changing all values in the tuple • Use the generalized projection operator to do this task r F ,F 1 2 , ,F l , (r ) • Each Fi is either – the i th attribute of r, if the ith attribute is not updated, or, – if the attribute is to be updated Fi is an expression, involving only constants and the attributes of r, which gives the new value for the attribute Update Examples • Make interest payments by increasing all balances by 5 percent. account account_number, branch_name, balance * 1.05 (account) Pay all accounts with balances over $10,000 6 percent interest and pay all others 5 percent account account_number, branch_name, balance * 1.06 ( BAL 10000 (account )) account_number, branch_name, balance * 1.05 (BAL 10000 (account)) Example Queries • Find the names of all customers who have a loan at the Perryridge branch. customer-name (branch-name=“Perryridge” (borrower.loan-number = loan.loan-number(borrower x loan))) • Find the names of all customers who have a loan at the Perryridge branch but do not have an account at any branch of the bank. customer-name (branch-name = “Perryridge” (borrower.loan-number = loan.loan-number(borrower x loan))) – customer-name(depositor) Example Queries • Find the largest account balance 1. Rename account relation as d 2. The query is: balance(account) - account.balance (account.balance < d.balance (account x d (account))) Example Queries • Find all customers who have an account from the “Downtown” and the Uptown” branches. Query 1 CN(BN=“Downtown”(depositor account)) CN(BN=“Uptown”(depositor account)) where CN denotes customer-name and BN denotes branch-name. Example Queries • Find all customers who have an account at all branches located in Brooklyn city. customer-name, branch-name (depositor account) branch-name (branch-city = “Brooklyn” (branch)) Exercises • • • • Employee(ename,str,city) Works(ename,cname,sal) Company(cname,city) Manages(ename,mname) Employee Joe Pine Mike Pine Carol Oak Kent Canton Kent Matt Lucy Sean Cleveland Kent Kent Main Pine Pine Manages Joe Lucy Mike Lucy Carol Matt Lucy Matt Sean Lucy Works Joe GE 30K Mike GE 100K Lucy GE 60K Sean GE 40K Carol GE 70K Matt GE 40K Company GE Cleveland IBM NYC Find names of employees that live in the same city and the same street as their managers • Employee (Employee Joe Pine Kent Mike Pine Canton Carol Oak Kent Lucy Pine Kent Sean Pine Kent Manages: Manages) Lucy Lucy Matt Matt Lucy Employee2 Where mname=employee2.ename & street =employee2.street & city=employee2.street Joe Pine Kent Mike Pine Canton Carol Oak Kent Lucy Pine Kent Sean Pine Kent Lucy Lucy Matt Matt Lucy Project on ename: Joe Sean Pine Kent Pine Kent Main Cleveland Main Cleveland Pine Kent Joe Pine Kent Lucy Pine Kent Sean Pine Kent Lucy Pine Kent Find Employees that make more than their managers • Works Manages: (Works Manages) Joe GE Mike GE Carol GE Lucy GE Sean GE 30K 100K 70K 60K 40K Lucy Lucy Matt Matt Lucy Works2 Where mname=works2.ename &salary >works2.salary Joe Mike Carol Lucy Sean GE 30K Lucy GE 60K GE 100K Lucy GE 60K GE 70K Matt GE 40K GE 60K Matt GE 40K GE 40K Lucy GE 60K Project on ename: Mike Carol Lucy Mike GE 100K Lucy GE 60K Carol GE 70K Matt GE 40K Lucy GE 60K Matt GE 40K Find all employees who make more money than any other employee Project Works on ename: Joe Lucy Sean Carol Mike Matt • Works Works2 Where sal<works2.sal Project on ename: Joe Lucy Sean Carol Matt Joe GE Joe GE Joe GE Joe GE Joe GE Lucy GE Lucy GE Sean GE Sean GE Sean GE Carol GE Matt GE Matt GE Matt GE 30K Mike GE 100K 30K Carol GE 70K 30K Lucy GE 60K 30K Sean GE 40K 30K Matt GE 40K 60K Carol GE 70K 60K Mike GE 100K 40K Mike GE 100K 40K Carol GE 70K 40K Lucy GE 60K 70K Mike GE 100K 40K Mike GE 100K 40K Carol GE 70K 40K Lucy GE 60K Substract from first projection the second one: Mike Find all employees that live in the same city as their company Works Company Employee Cname=company.cname & ename=employee.ename & city=works.city Matt GE 40K • Project on ename: Matt Cleveland Main Cleveland Extra Material Expression Trees Leaves are operands --- either variables standing for relations or particular relations Interior nodes are operators applied to their descendents customer-name, branch-name depositor account Relational Algebra on Bags • A bag is like a set but it allows elements to be repeated in a set. • Example: {1, 2, 1, 3, 2, 5, 2} is a bag. • Difference between a bag and a list is that order is not important in a bag. • Example: {1, 2, 1, 3, 2, 5, 2} and {1,1,2,3,2,2,5} is the same bag Need for Bags • SQL allows relations with repeated tuples. Thus SQL is not a relational algebra but rather “bag” algebra • In SQL one need to specifically ask to remove duplicates, otherwise replicated tuples will not be eliminated • Operation projection is more efficient on bags than on sets Operations on Bags • Select applies to each tuple and no duplicates are eliminated • Project also applies to each tuple and duplicates are not eliminated. Example A B C 1 2 3 1 2 5 2 3 7 Projection on A, B A 1 1 2 B 2 2 3 Other Bag Operations • An element in the union appears the number of times it appears in both bags • Example: {1, 2, 3, 1} UNION {1, 1, 2, 3, 4, 1} = {1, 1, 1, 1, 1, 2, 2, 3, 3, 4} • An element appears in the intersection of two bags is the minimum of the number of times it appears in either. • Example (con’t): {1, 2, 3, 1} INTERSECTION {1, 1, 2, 3, 4, 1} = {1, 1, 2, 3} • An element appears in the difference of two bags A and B as it appears in A minus the number of times it appears in B but never less that 0 times Bag Laws • Not all laws for set operations are valid for bags: • Commutative law for union does hold for bags: R UNION S = S UNION R • However S union S = S for sets and it is not equal to S if S is a bag • Examples Sailors Reserves sid bid day sid sn am e rat ing ag e 22 58 101 103 10/ 1 0/ 9 6 11/ 1 2/ 9 6 22 dus tin 7 45.0 31 lub b er 8 55.5 58 rusty 10 35.0 • Boats bid 101 102 103 104 b n am e Inte rlake Inte rlake C lip p er Ma rin e co lor B lue R ed Gr een R ed Find names of sailors who’ve reserved boat #103 • Solution 1: • Solution 2: snam e (( bid 103 R e serves ) Sailors ) snam e ( ( R e serves Sailors )) bid 103 Find names of sailors who’ve reserved a red boat • Information about boat color only available in Boats; so need an extra join: snam e (( B oats ) R e serves Sailors ) color ' red ' A more efficient (???) solution: sname ( (( ( Boats )) sid bid color 'red ' Re s) Sailors ) A query optimizer can find this given the first solution! Find sailors who’ve reserved a red or a green boat • Can identify all red or green boats, then find sailors who’ve reserved one of these boats: (T em pboats, ( color ' red ' color ' green ' B oats )) snam e (T em pboats R e serves Sailors ) Find sailors who’ve reserved a red and a green boat • Previous approach won’t work! Must identify sailors who’ve reserved red boats, sailors who’ve reserved green boats, then find the intersection (note that sid is a key for Sailors): (T em pred , sid (Tem pgreen, ( ( sid color ' red ' (( B oats ) R e serves )) color ' green ' B oats ) R e serves )) snam e ((T em pred T em pgreen ) Sailors ) Find the names of sailors who’ve reserved all boats • Uses division; schemas of the input relations to / must be carefully chosen: (T em psids, ( sid , bid R e serves ) / ( bid B oats )) snam e (T em psids Sailors ) To find sailors who’ve reserved all ‘Interlake’ boats: ..... / bid ( bnam e ' Interlake' B oats )

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# Relational Algebra